describe the end behavior of `f(x) = -3x^(38) + 7x^3 -12x + 13`

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Describe the end behavior of `f(x)=-3x^38+7x^3-12x+13` :

The end behavior of polynomials is fully determined by the first term, assuming it is written in standard form (i.e. the terms are written in descending degree)

The first term can have odd or even degree, and it can be positive or negative.

All polynomials of even degree (leading term of even degree) have the same type of end behavior: If the coefficient is positive, the function grows without bound as x goes to positive or negative infinity; while the function decreases without bound as x goes to positive or negative infinity.

All polynomials of odd degree (leading term of odd degree) have the same type of end behavior: If the leading coefficient is positive then the function increases on the entire domain -- in other words as x goes to negative infinity the function decreases without bound, and as x goes to positive infinity the function increases without bound. If the leading coefficient is negative the function decreases on the entire domain -- as x goes to negative infinity the function increases without bound, and as x goes to positive infinity the function decreases without bound.

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Here, f(x) has an even degreed first term with a negative coefficient.

`lim_(x->-oo)f(x)=-oo` or as x tends to negative infinity, the function decreases without bound.

`lim_(x->oo)f(x)=-oo` or as x tends to infinity the function decreases without bound.

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